COMPANION MATRIX FOR SUPERPOSITION OF POLYNOMIALS AND ITS APPLICATION TO KNOT THEORY

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The note provides a new formula for the companion matrix of the superposition of two polynomials over a commutative ring. The results obtained are used to provide a constructive proof of Plans’ theorem for two-bridge knots, which states that the first homology group of an odd-sheeted cyclic covering of a three-dimensional sphere branched over a given knot is the direct sum of two copies of some Abelian group. A similar result is also true for the homology of even-sheeted coverings factored by the reduced homology group of a two-sheeted covering. The structure of the above mentioned Abelian groups is described through Chebyshev polynomials of the second and fourth kind.

Sobre autores

A. Mednykh

Sobolev Institute of Mathematics; Novosibirsk State University

Email: smedn@mail.ru
Novosibirsk, Russia

I. Mednykh

Sobolev Institute of Mathematics; Novosibirsk State University

Email: ilyamednykh@mail.ru
Novosibirsk, Russia

G. Sokolova

Sobolev Institute of Mathematics; Novosibirsk State University; Novosibirsk State Technical University

Email: g.sokolova@g.nsu.ru
Novosibirsk, Russia

Bibliografia

  1. Bini D.A., Pan V.Y. Polynomial and Matrix Computations // Fundamental Algorithms. Birkhauser. Boston. MA. 1994. V. 1.
  2. Davis P.J. Circulant Matrices. New York: AMS Chelsea Publishing. 1994.
  3. Noferini V., Williams G. Matrices in companion rings, Smith forms, and the homology of 3-dimensional Brieskorn manifolds // J. Algebra. 2021. V 587. P. 1-19. https://doi.org/10.1016/j.jalgebra.2021.07.018
  4. Plans A. Aportacion al estudio de los grupos de homologia de los recubrimientos ciclicos ramificados correspondiente a un nudo // Rev. R. Acad. Cienc. Exactas, Fis. Nat. Madr. 1953. V. 47. P. 161-193.
  5. Del Val P., Weber C. Plans’ theorem for links // Topol. Appl. 1990. V. 34. P. 247-255. https://doi.org/10.1016/0166-8641(90)90041-Y
  6. Gordon C. McA. A short proof of a theorem of Plans on the homology of the branched cyclic coverings of a knot // Bull. Amer. Math. Soc. 1971. V. 168. P. 85-87. https://doi.org/10.1090/S0002-9904-1971-12611-3
  7. Stevens W.H. On the Homology of Branched Cyclic Covers of Knots // LSU Historical Dissertations and Theses. 1996. doi: 10.31390/gradschool_disstheses.6282
  8. Vaserstein L.N., Wheland E. Commutators and Companion Matrices over Rings of Stable Rank 1 // Linear Algebra Appl. 1990. V. 142. P. 263-277. doi: 10.1016/0024-3795(90)90270-M
  9. Brand L. The companion matrix and its properties // Amer. Math. Monthly. 1964. V. 71. N. 6. P. 629-634. doi: 10.1080/00029890.1964.11992294
  10. Lancaster P., Tismenetsky M. The Theory of Matrices. Second Edition with Applications. San Diego: Academic press. 1985.
  11. Mason J.C., Handscomb D.C. Chebyshev Polynomials. Boca Raton: CRC Press, 2003.
  12. Nakanishi Y., Suketa M. Alexander polynomials of two-bridge knots // J. Austral. Math. Soc. (Series A). 1996. V. 60. P. 334-342. https://doi.org/10.1017/S1446788700037848
  13. Murasugi K. On the Alexander polynomial of the alternating knot // Osaka J. Math. 1958. V. 10. P. 181-189.
  14. Hartley R.I. On two-bridged knot polynomials // J. Austral. Math. Soc. (Series A). 1979. V. 28. P. 241-249. https://doi.org/10.1017/S1446788700015743
  15. Schubert H. Knoten mit zwei Bracken // Math. Z. 1956. V. 65. V. P. 133-170.
  16. Cattabriga A., Mulazzani M. Strongly-cyclic branched coverings of (1, 1)-knots and cyclic presentations of groups // Math. Proc. Cambridge Phil. Soc. 2003. V. 135. N. 1. P. 137-146. https://doi.org/10.1017/S0305004103006686
  17. Fox R.H. Free Differential Calculus III. Subgroups // Ann. Math. 1956. V. 64 N. 3. P. 407-419.
  18. Mednykh I.A. Homology group of branched cyclic covering over a 2-bridge knot of genus two. Preprint. 2021. arXiv:2111.04292 [math.CO].
  19. Seifert H. Uber das Geschlecht von Knoten // Math. Ann. 1934. V. 110. P. 571-592.
  20. Kutateladze S.S. Fundamentals of functional analysis. Netherlands: Springer Science and Business Media, 2013.
  21. Reidemeister K. Knotentheorie. New York: Chelsea Pub. Co., NewYork, 1948.
  22. Raymond Lickorish W.B. An Introduction to Knot Theory. New York: Springer, 1997.

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